here is a (legitimate link to a) reference that purports to treat such questions of relative size of gadgets used in category theory:
http://katmat.math.uni-bremen.de/acc/acc.pdf
".............
Hence sets are special classes. Classes that are not sets are called proper classes. They cannot be members of any class. Because of this, Russell’s paradox now translates into the harmless statement that the class of all sets that are not members of themselves is a proper class. Also the universe U, the class of all vector spaces, the class of all topological spaces, and the class of all automata are proper classes.
Notice that in this setting condition 2.1(4)(a) above gives us the Axiom of Replacement : (5) there is no surjection from a set to a proper class.
This means that each set must have “fewer” elements than any proper class.
Therefore sets are also called small classes, and proper classes are called large classes. This distinction between “large” and “small” turns out to be crucial for many categorical considerations.5
The framework of sets and classes described so far suffices for defining and investigating such entities as the category of sets, the category of vector spaces, the category of topological spaces, the category of automata, functors between these categories, and natural transformations between such functors. Thus for most of this book we need not go beyond this stage. Therefore we advise the beginner to skip from here, go directly to §3, and return to this section only when the need arises.
The limitations of the framework described above become apparent when we try to per- form certain constructions with categories; e.g., when forming “extensions” of categories or when forming categories that have categories or functors as objects. Since members of classes must be sets and U is not a set, we can’t even form a class {U} whose only member is U, much less a class whose members are all the subclasses of U or all functions from U to U. In order to deal effectively with such “collections” we need a further level of generality:
2.3 CONGLOMERATES
The concept of “conglomerate” has been created to deal with “collections of classes”. In particular, we require that:
- (1) every class is a conglomerate,
- (2) for every “property” P , one can form the conglomerate of all classes with property
P,
- (3) conglomerates are closed under analogues of the usual set-theoretic constructions outlined above (2.1); i.e., they are closed under the formation of pairs, unions, products (of conglomerate-indexed families), etc."
So it has a veeery special place in my heart. Somehow it didn't contribute to my depression.